this paper gives a short survey of some of the known results generalizing the theorem, credited to i. schur, that if the central factor group is finite then the derived subgroup is also finite.

a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the...

a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the center of such...

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C∗-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we es...

We adapt Sarason’s proof of the Julia-Caratheodory theorem to the class of Schur-Agler mappings of the unit ball, obtaining a strengthened form of this theorem. In particular those quantities which appear in the classical theorem and depend only on the component of the mapping in the complex normal direction have K-limits (not just restricted K-limits) at the boundary. Let B denote the open uni...

We consider a more general class than the class of Schur multipliers namely the Abel-Schur multipliers, which in turn coincide with the bounded linear operators on `2 preserving the diagonals. We extend to the matrix framework Theorem 2.4 (a) of a paper of Anderson, Clunie, and Pommerenke published in 1974, and as an application of this theorem we obtain a new proof of the necessity of an old t...

When N is a normal subgroup of G, can we reconstruct G from N and G/N? In general, no. For instance, the groups Z/(p2) and Z/(p) × Z/(p) (for prime p) are nonisomorphic, but each has a cyclic subgroup of order p and the quotient by it also has order p. As another example, the nonisomorphic groups Z/(2p) and Dp (for odd prime p) have a normal subgroup that is cyclic of order p, whose quotient is...

The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite co...